Elasticity/Torsion of noncircular cylinders

Torsion of Non-Circular Cylinders

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Torsion of a noncircular cylinder

About the problem

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  • Solution first found by St. Venant.
  • Tractions at the ends are statically equivalent to equal and opposite torques  .
  • Lateral surfaces are traction-free.

Assumptions:

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  • An axis passes through the center of twist (  axis).
  • Each c.s. projection on to the   plane rotates,but remains undistorted.
  • The rotation of each c.s. ( ) is proportional to  .
 

where   is the twist per unit length.

  • The out-of-plane distortion (warping) is the same for each c.s. and is proportional to  .

Find:

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  • Torsional rigidity ( ).
  • Maximum shear stress.

Solution:

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Displacements

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where   is the warping function.

If   (small strain),

 

Strains

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Therefore,

 

Stresses

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Therefore,

 

Equilibrium

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Therefore,

 

Internal Tractions

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  • Normal to cross sections is  .
  • Normal traction  .
  • Projected shear traction is  .
  • Traction vector at a point in the cross section is tangent to the cross section.

Boundary Conditions on Lateral Surfaces

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  • Lateral surface traction-free.
  • Unit normal to lateral surface appears as an in-plane unit normal to the boundary  .

We parameterize the boundary curve   using

 

The tangent vector to   is

 

The tractions   and   on the lateral surface are identically zero. However, to satisfy the BC  , we need

 

or,

 

Boundary Conditions on End Surfaces

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The traction distribution is statically equivalent to the torque  . At  ,

 

Therefore,

 

From equilibrium,

 

Hence,

 

The Green-Riemann Theorem

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If   and   then

 

with the integration direction such that   is to the left.

Applying the Green-Riemann theorem to equation (17), and using equation (16)

 

Similarly, we can show that  .   since  .

The moments about the   and   axes are also zero.

The moment about the   axis is

 

where   is the torsion constant. Since  , we have

 

If  , then  , the polar moment of inertia.

Summary of the solution approach

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  • Find a warping function   that is harmonic. and satisfies the traction BCs.
  • Compatibility is not an issue since we start with displacements.
  • The problem is independent of applied torque and the material properties of the cylinder.
  • So it is just a geometrical problem. Once   is known, we can calculate
    • The displacement field.
    • The stress field.
    • The twist per unit length.